ByGodfrey Sperling Jr., Staff correspondent of The Christian Science Monitor, Godfrey Sperling, the Monitor's Washington bureau chief, has held breakfasts for reporters featuring key national and international figures for the past 16 years.February 7, 1983

Washington
— The President, brave front to the contrary, is not without his anxieties. The recovery isn't moving fast enough. The Israelis continue to obstruct his Mideast peace initiative. The Democrats are emboldened by recent election victories. . . .

Not the least of the President's concerns is that too many women in America just don't seem to like him - a solid majority according to several respected polls. Mr. Reagan and his cohorts are puzzled - but they know they have a problem.

''I believe the gender gap is real,'' says the new GOP national chairman, Frank J. Fahrenkopf Jr. He calls it a ''perception problem,'' and one that is ''particularly serious'' for the Republican Party among women, age 21 to 40, who are either single or single heads of households.

''The perception,'' concedes the chairman, in office only a few days, ''is that this administration doesn't care about (these women), and that this President is going to make life more difficult for them through his economic programs.''

The President has been trying particularly hard of late to tell women that he has their interests at heart. First, he elevated Elizabeth Hanford Dole to the Cabinet. Then he chose Margaret M. Heckler for a Cabinet post. This was a reminder to many that it was Reagan who had placed the first woman, Sandra Day O'Connor, on the US Supreme Court.

But what might Mr. Fahrenkopf say of such moves? Well, he was telling a group of reporters that high-level appointments of blacks to the Reagan administration could not be the answer to what he conceded was the Republicans' inability to attract blacks to the Reagan cause. ''Those are only symbols,'' he said.

What was needed, he said, was for blacks to see that they must broaden their power base to include both parties. He said that they should see it is in their best interests to participate in the Republican Party - so that they can be a part of Republican leadership when that party is in power, and so that the Democrats aren't in a position always to take them for granted.

Fahrenkopf said that he is going to make an all-out effort to bring all minorities into participation in party activities, from the grass roots up.

But for women he has a very special project in the blueprint stage. He plans to hold, under GOP auspices, a series of meetings across the country with what he calls a broad cross section of women - ''not just Republican women.''

From these meetings, he said, ''we hope to hear what their real concerns are.'' Furthermore, he added, these meetings ''will show that we really do care, and that we are going to make the effort to respond.''

The questioning from reporters that followed this announcement seemed to indicate that the news media are less than convinced that the Republican Party can make much ground with such meetings.

When do you plan to begin this?

I hope we would start by late spring. It's a top priority with me.

How are you going to overcome the feeling among women that the Republicans dumped a pro-ERA plan that had been put in the platform by Republicans even more conservative than Ronald Reagan?

You can't say the women of this country are solidly behind the Equal Rights Amendment.

No, I say you alienated a large body of women who were for it.

The committee made that change. Obviously the majority of that committee believed that changes aiding women's rights belonged with the state legislatures - because of progress being made there.

Fahrenkopf maintained his aplomb - and his enthusiasm for his women-related program. He said that after hearing what women's concerns are, ''we'll see if we can't come up with some programs that we can offer as party proposals . . . to the administration.''

Can this Fahrenkopf initiative help the GOP close the gender gap? It might. And, at least, it shows that Republicans see that they must do more to make more women feel that Mr. Reagan is their President and that he really cares for them.

The Fahrenkopf plan for meetings with women, which opponents are certain to brand a ''political gimmick,'' shows something more, too: It is just another sign of a President who, after two years in office, is viewing his position in a realistic way - and making some concessions.